Question: Simplify and expand the following expression: $ \dfrac{5q}{q - 6}-\dfrac{3q}{5q + 6} $
Answer: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(q - 6)(5q + 6)$ Multiply the first term by $\dfrac{5q + 6}{5q + 6}$ $ \begin{align*} \dfrac{5q}{q - 6} \times \dfrac{5q + 6}{5q + 6} & = \dfrac{(5q)(5q + 6)}{(q - 6)(5q + 6)} \\ & = \dfrac{25q^2 + 30q}{(q - 6)(5q + 6)}\end{align*} $ Multiply the second term by $\dfrac{q - 6}{q - 6}$ $ \begin{align*} \dfrac{3q}{5q + 6} \times \dfrac{q - 6}{q - 6} & = \dfrac{(3q)(q - 6)}{(5q + 6)(q - 6)} \\ & = \dfrac{3q^2 - 18q}{(5q + 6)(q - 6)}\end{align*} $ Now we have: $ = \dfrac{25q^2 + 30q}{(q - 6)(5q + 6)} - \dfrac{3q^2 - 18q}{(5q + 6)(q - 6)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{25q^2 + 30q - (3q^2 - 18q)}{(q - 6)(5q + 6)} $ $ = \dfrac{25q^2 + 30q - 3q^2 + 18q}{(q - 6)(5q + 6)} $ $ = \dfrac{22q^2 + 48q}{(q - 6)(5q + 6)}$ Expand the denominator: $ = \dfrac{22q^2 + 48q}{5q^2 - 24q - 36}$